A note on the string topology BV-algebra for $S^2$ with $Z_2$ coefficients

TL;DR

This paper explores the relationship between string topology BV-algebras and Hochschild cohomology for $S^2$ with $Z_2$ coefficients, demonstrating how higher homotopies induce Poincaré duality structures at the cochain level.

Contribution

It introduces a method to obtain the string topology BV algebra on Hochschild cohomology via Poincaré duality with higher homotopies at the cochain level.

Findings

String topology BV algebra can be derived from Hochschild cohomology using higher homotopy Poincaré duality.

Higher homotopies induce a Poincaré duality structure on cohomology from the cochain level.

The approach clarifies the non-isomorphism between the two BV algebra structures.

Abstract

Luc Menichi showed that the BV algebras on $H^\bullet(LS^2;Z_2)[-2]$ coming from string topology and the one on $HH^\bullet(H^\bullet(S^2;Z_2),H^\bullet(S^2;Z_2))$ using Poincar\'e duality on $H^\bullet(S^2;Z_2)$ are not isomorphic. In this note we show how one can obtain the string topology BV algebra on Hochschild cohomology using a Poincar\'e duality structure with higher homotopies. This Poincar\'e duality (with higher homotopies) on cohomology is induced by a local Poincar\'e duality (with higher homotopies) on the cochain level.